De Jong's late goal, his sixth in the league for Barca, proved the decisive moment in a 3-2 victory in Valencia but it was the introduction of the 19-year-old Pedri off the bench that turned the momentum of the contest.
A frantic 11 minutes early in the second half saw Levante score one penalty in the 52nd minute through Jose Luis Morales before missing a second in the 55th, this time Roger Marti stepping up, only to be denied by Marc-Andre ter Stegen.
Pedri came off the bench in the 56th minute and three minutes later Barcelona equalised, Pierre-Emerick Aubameyang heading in Ousmane Dembele's cross to score his 10th goal for his new club.
Then in the 63rd minute, Pedri fired Barca in front with a first-time finish into the corner, only for Levante to level again after being awarded a third penalty, Gonzalo Melero this time making no mistake.
Levante looked set to claim a point but De Jong snatched the win, meeting Jordi Alba's cross and heading in at the near post.
Barca climb back above Sevilla into second, still 12 points behind Real Madrid, who eased to a 2-0 victory over Getafe on Saturday.
Xavi Hernandez's side have a game in hand over Real Madrid, Sevilla and Atletico Madrid, with Atletico three points behind in fourth after losing on Saturday away at Mallorca.
Pedri scored a brilliant winner against Sevilla last weekend and while the midfielder was initially rested against Levante, Barcelona were transformed after he came on.
Aubameyang, meanwhile, continued his own rich vein of form since leaving Arsenal in January. The Gabon striker now has 10 goals in 14 games for his new club.
Levante were all over Barcelona early on and Morales was thwarted by a excellent, goal-line clearance by Eric Garcia after his finish at the end of a weaving run looked to be spinning in. Garcia scrambled back to hook away.
Ferran Torres twice went close but the mayhem began shortly after halftime when Dani Alves leant into Son to prevent him retrieving a one-two and Morales scored from the spot.
Three minutes later, Levante had another penalty after the ball cruelly dropped down onto the arm of Garcia. Morales left this spot-kick to Roger but his low shot to the left was palmed away by Ter Stegen.
Levante knew an opportunity had been squandered and Barca played quicker with Pedri on the pitch, the teenager playing the key pass out to Dembele, who crossed in for Aubameyang to nod in.
Five minutes later, Barcelona were in front as Gavi nipped between two Levante defenders and squared for Pedri, who bent the ball first time into the far corner.
Levante, though, came again and when Melero fired in their third penalty of the night after Dani Gomez was clipped by Clement Lenglet, they looked to have secured a point.
De Jong, on for Aubameyang, had other ideas, dashing into the area in the 92nd minute and Alba's curving cross finding him. De Jong headed the ball inside the post.